A few remarks
The picture is misleading and ingores that both antennas, the antenna providing the test signal for measurement and the antenna under test, have to meet the farfield conditions ( see Near and far field - Wikipedia ). The reqired separation distance to meet the far field requirements varies with the wavelength and antenna dimensions.
When the distance separtion is met the actual attenuation VNA signal may be to weak for measurement plus other problems are
the VNA will also receive all reflections, which will vary with polarisation, antenna height and ground reflectivity
and receive also other signals that are radiated
Also keep in mind that since you radiate the VNA signal you most likely generate interference for other systems, since the signals of the VNA are normally shielded by coaxial cables and not radiated via an antenna. For this reason comercial antenna tests do not use UWB signals.
You can find more information if you look e.g. for measurement of RADAR antenna
